In high-speed wireless communications, multiple streams are ideally transmitted over the air simultaneously. For example, a Multiple Input Multiple Output (MIMO) scheme enables multiple-stream transmission by using multiple transmits and/or receive antennas. Each stream is called a layer and corresponds to a sequence of modulation symbols. The layers are either directly mapped to the antenna ports or may first undergo a transformation by means of channel dependent or channel independent pre-coding.
The multiple streams come from single/multiple code words. The code words should be mapped into multiple streams in a proper way. If multiple streams experience different channel quality, it is often beneficial to shift the layer mapped from a specific code word by a certain amount, which effectively transmits each code word across multiple layers. This layer shifting helps achieve diversity gain which leads to improved error performance. The layer shifting is generally performed in a per-sample basis to maximize diversity gain. Thus, the layer or layers mapped from a code word are shifted for every sample (i.e., every modulation symbol).
To decompose the multiple layers at the receiver, several equalization techniques may be used. The simplest equalization is linear equalization, such as Zero Forcing (ZF)/Minimum Mean Square Error (MMSE) equalization. To improve the error performance, decision feedback equalization called SIC may be used when multiple code words are transmitted. Since SIC cancels the previously decoded layers from the received signal, SIC always outperforms linear equalization if the previous decoding is sufficiently reliable. One of the disadvantages of SIC is that SIC requires additional computation, such as signal reconstruction and cancellation.
Oftentimes, layer shifting is applied to a multiple layer MIMO transmission combined with Discreet Fourier Transform Spread (DFTS)-Orthogonal Frequency Division Multiplexing (OFDM). DFTS-OFDM is a technique for single-carrier transmission used in a Long Term Evolution (LTE) uplink for at least single antenna transmissions. In each subframe, spanning 1 millisecond, transmission of a block of information bits corresponding to a transport block is possible. FIG. 1 is a simplified block diagram of components of an existing DFTS-OFDM system. The system includes a modulator 10, a DFT precoder 12, an Inverse Discrete Fourier Transform (IDFT) 14, and a cyclic prefix adder 16 for processing a signal 18. The information block is coded and passed through the modulator 10 to produce information carrying symbols that are subsequently transformed to the frequency domain by means of the Discreet Fourier Transform (DFT) precoder. The output of the DFT precoder is connected to potentially only a subset of the inputs of the IDFT 14, producing a time domain signal corresponding to one DFTS-OFDM symbol. A cyclic prefix is added (at the cyclic prefix adder 16) to the time domain signal which is thereafter transmitted over the channel. This enables transmissions over only a part of the system bandwidth while simplifying channel equalization in the receiver by utilizing an efficient support of frequency domain equalization.
For an example, a subframe is divided into two slots and comprises a number of DFTS-OFDM symbols and two symbols for reference signals. The number of DFTS-OFDM symbols depends on the length of the cyclic prefix and is either 12 or 10.
As mentioned previously, linear equalization is simple, but suffers from nontrivial performance loss. One attractive solution is decision feedback equalization, such as SIC. However, when per-sample layer shifting is applied to DFTS-OFDM, the equalization following each cancellation becomes extremely complex since the equivalent system after cancellation is no longer seen as the per-layer DFT followed by a per-subcarrier frequency-flat fading channel. Thus, the equalization may be performed over the entire DFTS-OFDM symbol, which results in prohibitively high computational complexity (as opposed to multi-carrier systems or single carrier systems without per-sample layer shifting whose equalization is performed on a per-subcarrier basis).
To illustrate this problem, assume that there are Nt transmit antennas and two receive antennas, and a DFTS-OFDM symbol consists of two layers and K subcarriers. The 2K×1 vector symbol y representing the frequency-domain received symbols within a DFTS-OFDM symbol is then expressed asy=√{square root over (αK)}E(FKI2)s+w  (1)where
FK is a K×K matrix representing the DFT,
E is a 2K×2K matrix representing the equivalent channel (including the precoder),
s is a 2K×1 vector representing the transmitted symbols,
w is a 2K×1 vector representing the noise symbols.
FKI2 in (1) represents the per-layer DFT. In addition, the transmit power is normalized through α, which is defined as
      α    =                  N        K            ⁢                        E          s                          N          t                      ,where
N is the total number of subcarriers,
Es is the received energy per subcarrier.
Since each subcarrier experiences frequency-flat fading due to orthogonality, the equivalent channel matrix E is a block-diagonal matrix expressed asE=diag(E0, E1, . . . , EK−1)  (2)where the 2×2 matrix Ek represents the equivalent channel (including the precoder) at the k-th subcarrier.
For illustrative purposes, assume that two code words are mapped into two layers according to per-sample layer shifting. FIG. 2 is a simplified block diagram illustrating layer shifting in a per-sample basis in an existing system. Specifically, code word (CW) #1 20 is mapped into a Layer #1 22 at the first sample. However, CW #1 is mapped into a Layer #2 24 at the second sample and vice versa. CW #2 26 is mapped at a first sample into the Layer #1 while the second sample is mapped into the Layer #2 and so forth. A sample corresponds to a time instant. In FIG. 2, each codeword spans K samples and two modulation symbols (one per codeword) are fed into a CW-to-layer mapping 30 at each time instant.
If CW #1 is decoded first, the 2K×1 vector symbol y′ representing the frequency-domain received symbols after cancellation is expressed asy′=√{square root over (αK)}Gs′+w  (3)where
G is 2K×K vector representing the equivalent channel (including the precoder and DFT),
s′ is a K×1 vector representing the transmitted symbols after cancellation.
Here G is given as
                    G        =                  (                                                                                          e                                          0                      ,                      1                                                        ⁢                                      f                                          0                      ,                      0                                                                                                                                        e                                          0                      ,                      0                                                        ⁢                                      f                                          0                      ,                      1                                                                                                                                        e                                          0                      ,                      1                                                        ⁢                                      f                                          0                      ,                      2                                                                                                                                        e                                          0                      ,                      0                                                        ⁢                                      f                                          0                      ,                      3                                                                                                  …                                                                                  e                                          0                      ,                      1                                                        ⁢                                      f                                          0                      ,                                              K                        -                        2                                                                                                                                                              e                                          0                      ,                      0                                                        ⁢                                      f                                          0                      ,                                              K                        -                        1                                                                                                                                                                                      e                                          1                      ,                      1                                                        ⁢                                      f                                          1                      ,                      0                                                                                                                                        e                                          1                      ,                      0                                                        ⁢                                      f                                          1                      ,                      1                                                                                                                                        e                                          1                      ,                      1                                                        ⁢                                      f                                          1                      ,                      2                                                                                                                                        e                                          1                      ,                      0                                                        ⁢                                      f                                          1                      ,                      3                                                                                                  …                                                                                  e                                          1                      ,                      1                                                        ⁢                                      f                                          1                      ,                                              K                        -                        2                                                                                                                                                              e                                          1                      ,                      0                                                        ⁢                                      f                                          1                      ,                                              K                        -                        1                                                                                                                                                ⋮                                            ⋮                                            ⋮                                            ⋮                                            ⋱                                            ⋮                                            ⋮                                                                                                          e                                                                  K                        -                        1                                            ,                      1                                                        ⁢                                      f                                                                  K                        -                        1                                            ,                      0                                                                                                                                        e                                                                  K                        -                        1                                            ,                      0                                                        ⁢                                      f                                                                  K                        -                        1                                            ,                      1                                                                                                                                        e                                                                  K                        -                        1                                            ,                      1                                                        ⁢                                      f                                                                  K                        -                        1                                            ,                      2                                                                                                                                        e                                                                  K                        -                        1                                            ,                      0                                                        ⁢                                      f                                                                  K                        -                        1                                            ,                      3                                                                                                  …                                                                                  e                                                                  K                        -                        1                                            ,                      1                                                        ⁢                                      f                                                                  K                        -                        1                                            ,                                              K                        -                        2                                                                                                                                                              e                                                                  K                        -                        1                                            ,                      0                                                        ⁢                                      f                                                                  K                        -                        1                                            ,                                              K                        -                        1                                                                                                                          )                                    (        4        )            where
ek,n is the n-th column vector of Ek,
fk,l is the (k,l)-entry of FK.
In order to decompose G into a block-diagonal matrix (representing the per-subcarrier frequency-flat fading channel) and a unitary matrix (representing the per-layer DFT), each row of G should consist of block matrices which are given by scalar multiplications of the same matrix. However, as shown in equation (4), the decomposition is impossible because of layer shifting within a DFTS-OFDM symbol. For example, the sub-matrices at the k-th row of G are given by scalar multiplications of either ek,0 or ek,1 (depending on the column indices). Therefore, it would be advantageous to perform equalization over the entire DFTS-OFDM symbol (neither in a per-subcarrier basis nor in a per-layer basis as opposed to that without layer shifting), which causes extremely high computational complexity.
Layer shifting within a DFTS-OFDM symbol is undesirable for other reasons. Consider the matrix X of frequency domain 2×1 transmit vectors after DFT precoding:X=[s0 s1 . . . sK−1]FKT=SFKT x=vec(X)=vec(I2SFKT)={vec(ABC)=(CTA)vec(B)}=(FKI2)vec(S)=[x0(1) x0(2) x1(1) x1(2) x2(1)x2(2) . . . xK−1(1) xK−1(2)]T The received signal vector y can then be written as
                    y        =                ⁢                              [                                                                                E                    0                                                                    0                                                  …                                                  0                                                                              0                                                                      E                    1                                                                                                                                                            ⋮                                                                              ⋮                                                                                                                                          ⋱                                                  0                                                                              0                                                  …                                                  0                                                                      E                                          K                      -                      1                                                                                            ]                    ⁢          x                                        =                ⁢                              [                                                                                E                    0                                                                    0                                                  …                                                  0                                                                              0                                                                      E                    1                                                                                                                                                            ⋮                                                                              ⋮                                                                                                                                          ⋱                                                  0                                                                              0                                                  …                                                  0                                                                      E                                          K                      -                      1                                                                                            ]                    ⁢                      (                                          F                K                            ⊗                              I                2                                      )                    ⁢                      vec            ⁡                          (              S              )                                                              =                ⁢                              [                                                                                E                    0                                                                    0                                                  …                                                  0                                                                              0                                                                      E                    1                                                                                                                                                            ⋮                                                                              ⋮                                                                                                                                          ⋱                                                  0                                                                              0                                                  …                                                  0                                                                      E                                          K                      -                      1                                                                                            ]                    ⁢                                    (                                                F                  K                                ⊗                                  I                  2                                            )                        ⁡                          [                                                                                          s                      0                                              (                        1                        )                                                                                                                                                        s                      0                                              (                        2                        )                                                                                                                                                        s                      1                                              (                        1                        )                                                                                                                                                        s                      1                                              (                        2                        )                                                                                                                                  ⋮                                                                                                              s                                              K                        -                        1                                                                    (                        1                        )                                                                                                                                                        s                                              K                        -                        1                                                                    (                        2                        )                                                                                                        ]                                                              =                ⁢                              [                                                                                E                    0                                                                    0                                                  …                                                  0                                                                              0                                                                      E                    1                                                                                                                                                            ⋮                                                                              ⋮                                                                                                                                          ⋱                                                  0                                                                              0                                                  …                                                  0                                                                      E                                          K                      -                      1                                                                                            ]                    ⁢                                    (                                                (                                                            F                      K                                        ⊗                                          [                                                                                                    1                                                                                0                                                                                                                                0                                                                                0                                                                                              ]                                                        )                                +                                  (                                                            F                      K                                        ⊗                                          [                                                                                                    0                                                                                0                                                                                                                                0                                                                                1                                                                                              ]                                                        )                                            )                        ⁡                          [                                                                                          s                      0                                              (                        1                        )                                                                                                                                                        s                      0                                              (                        2                        )                                                                                                                                                        s                      1                                              (                        1                        )                                                                                                                                                        s                      1                                              (                        2                        )                                                                                                                                  ⋮                                                                                                              s                                              K                        -                        1                                                                    (                        1                        )                                                                                                                                                        s                                              K                        -                        1                                                                    (                        2                        )                                                                                                        ]                                                              =                ⁢                  (                                                    [                                                                                                    e                        0                                                  (                          1                          )                                                                                                            0                                                              …                                                              0                                                                                                  0                                                                                      e                        1                                                  (                          1                          )                                                                                                                                                                                                                        ⋮                                                                                                  ⋮                                                                                                                                                                          ⋱                                                              0                                                                                                  0                                                              …                                                              0                                                                                      e                                                  K                          -                          1                                                                          (                          1                          )                                                                                                                    ]                            ⁢                              (                                                      F                    K                                    ⊗                                      [                                                                                            1                                                                          0                                                                                      ]                                                  )                                      +                                                                      ⁢                                    [                                                                                          e                      0                                              (                        2                        )                                                                                                  0                                                        …                                                        0                                                                                        0                                                                              e                      1                                              (                        2                        )                                                                                                                                                                                                    ⋮                                                                                        ⋮                                                                                                                                                          ⋱                                                        0                                                                                        0                                                        …                                                        0                                                                              e                                              K                        -                        1                                                                    (                        2                        )                                                                                                        ]                        ⁢                          (                                                F                  K                                ⊗                                  [                                                                                    0                                                                    1                                                                              ]                                            )                                )                ⁡                  [                                                                      s                  0                                      (                    1                    )                                                                                                                        s                  0                                      (                    2                    )                                                                                                                        s                  1                                      (                    1                    )                                                                                                                        s                  1                                      (                    2                    )                                                                                                      ⋮                                                                                      s                                      K                    -                    1                                                        (                    1                    )                                                                                                                        s                                      K                    -                    1                                                        (                    2                    )                                                                                ]                                        =                ⁢                                            [                                                                                          e                      0                                              (                        1                        )                                                                                                  0                                                        …                                                        0                                                                                        0                                                                              e                      1                                              (                        1                        )                                                                                                                                                                                                    ⋮                                                                                        ⋮                                                                                                                                                          ⋱                                                        0                                                                                        0                                                        …                                                        0                                                                              e                                              K                        -                        1                                                                    (                        1                        )                                                                                                        ]                        ⁢                                          F                K                            ⁡                              [                                                                                                    s                        0                                                  (                          1                          )                                                                                                                                                                        s                        1                                                  (                          1                          )                                                                                                                                                ⋮                                                                                                                          s                                                  K                          -                          1                                                                          (                          1                          )                                                                                                                    ]                                              +                                                ⁢                              [                                                                                e                    0                                          (                      2                      )                                                                                        0                                                  …                                                  0                                                                              0                                                                      e                    1                                          (                      2                      )                                                                                                                                                                                ⋮                                                                              ⋮                                                                                                                                          ⋱                                                  0                                                                              0                                                  …                                                  0                                                                      e                                          K                      -                      1                                                              (                      2                      )                                                                                            ]                    ⁢                                    F              K                        ⁡                          [                                                                                          s                      0                                              (                        2                        )                                                                                                                                                        s                      1                                              (                        2                        )                                                                                                                                  ⋮                                                                                                              s                                              K                        -                        1                                                                    (                        2                        )                                                                                                        ]                                          
From the expression for the last equality sign, it is clear that each of the two terms has the structure block diagonal matrix multiplied by FFT matrix. Thus, frequency domain equalization followed by IFFT can be applied to decode either of the terms when the other term has been cancelled. SIC then achieves reasonable complexity. The last equation is consistent with equation (6) discussed below.
Considering the above expression with the layer shifting case having the DFTS-OFDM symbol:
                    y        =                ⁢                                            [                                                                                          e                      0                                              (                        1                        )                                                                                                  0                                                        …                                                        0                                                                                        0                                                                              e                      1                                              (                        1                        )                                                                                                                                                                                                    ⋮                                                                                        ⋮                                                                                                                                                          ⋱                                                        0                                                                                        0                                                        …                                                        0                                                                              e                                              K                        -                        1                                                                    (                        1                        )                                                                                                        ]                        ⁢                                          F                K                            ⁡                              [                                                                                                    s                        0                                                  (                          1                          )                                                                                                                                                                        s                        1                                                  (                          1                          )                                                                                                                                                ⋮                                                                                                                          s                                                  K                          -                          1                                                                          (                          1                          )                                                                                                                    ]                                              +                    ⁢                                    [                                                                                          e                      0                                              (                        2                        )                                                                                                  0                                                        …                                                        0                                                                                        0                                                                              e                      1                                              (                        2                        )                                                                                                                                                                                                    ⋮                                                                                        ⋮                                                                                                                                                          ⋱                                                        0                                                                                        0                                                        …                                                        0                                                                              e                                              K                        -                        1                                                                    (                        2                        )                                                                                                        ]                        ⁢                                          F                K                            ⁡                              [                                                                                                    s                        0                                                  (                          2                          )                                                                                                                                                                        s                        1                                                  (                          2                          )                                                                                                                                                ⋮                                                                                                                          s                                                  K                          -                          1                                                                          (                          2                          )                                                                                                                    ]                                                                            =                ⁢                                            [                                                                                                                  e                        0                                                  (                          1                          )                                                                    ⁢                                              f                                                  1                          ,                          1                                                                                                                                                                        e                        0                                                  (                          1                          )                                                                    ⁢                                              f                                                  1                          ,                          2                                                                                                                          …                                                                                                      e                        0                                                  (                          1                          )                                                                    ⁢                                              f                                                  1                          ,                          K                                                                                                                                                                                                        e                        1                                                  (                          1                          )                                                                    ⁢                                              f                                                  2                          ,                          1                                                                                                                                                                        e                        1                                                  (                          1                          )                                                                    ⁢                                              f                                                  2                          ,                          2                                                                                                                          …                                                                                                      e                        1                                                  (                          1                          )                                                                    ⁢                                              f                                                  2                          ,                          K                                                                                                                                                          ⋮                                                        ⋮                                                        ⋱                                                        ⋮                                                                                                                                      e                                                  K                          -                          1                                                                          (                          1                          )                                                                    ⁢                                              f                                                  K                          ,                          1                                                                                                                                                                        e                                                  K                          -                          1                                                                          (                          1                          )                                                                    ⁢                                              f                                                  K                          ,                          2                                                                                                                          …                                                                                                      e                                                  K                          -                          1                                                                          (                          1                          )                                                                    ⁢                                              f                                                  K                          ,                                                                                                                                ]                        ⁡                          [                                                                                          s                      0                                              (                        1                        )                                                                                                                                                        s                      1                                              (                        1                        )                                                                                                                                  ⋮                                                                                                              s                                              K                        -                        1                                                                    (                        1                        )                                                                                                        ]                                +                                    [                                                                                                                  e                        0                                                  (                          2                          )                                                                    ⁢                                              f                                                  1                          ,                          1                                                                                                                                                                        e                        0                                                  (                          2                          )                                                                    ⁢                                              f                                                  1                          ,                          2                                                                                                                          …                                                                                                      e                        0                                                  (                          2                          )                                                                    ⁢                                              f                                                  1                          ,                          K                                                                                                                                                                                                        e                        1                                                  (                          2                          )                                                                    ⁢                                              f                                                  2                          ,                          1                                                                                                                                                                        e                        1                                                  (                          2                          )                                                                    ⁢                                              f                                                  2                          ,                          2                                                                                                                          …                                                                                                      e                        1                                                  (                          2                          )                                                                    ⁢                                              f                                                  2                          ,                          K                                                                                                                                                          ⋮                                                        ⋮                                                        ⋱                                                        ⋮                                                                                                                                      e                                                  K                          -                          1                                                                          (                          2                          )                                                                    ⁢                                              f                                                  K                          ,                          1                                                                                                                                                                        e                                                  K                          -                          1                                                                          (                          2                          )                                                                    ⁢                                              f                                                  K                          ,                          2                                                                                                                          …                                                                                                      e                                                  K                          -                          1                                                                          (                          2                          )                                                                    ⁢                                              f                                                  K                          ,                          K                                                                                                                                ]                        ⁡                          [                                                                                          s                      0                                              (                        2                        )                                                                                                                                                        s                      1                                              (                        2                        )                                                                                                                                  ⋮                                                                                                              s                                              K                        -                        1                                                                    (                        2                        )                                                                                                        ]                                                              =                ⁢                  {                                                                      assume                  ⁢                                                                          ⁢                  second                  ⁢                                                                          ⁢                  codeword                  ⁢                                                                          ⁢                  is                  ⁢                                                                          ⁢                                      zero                    /                    canceled                                    ⁢                                                                          ⁢                  and                  ⁢                                                                          ⁢                  layer                  ⁢                                                                          ⁢                  shifting                  ⁢                                                                          ⁢                  is                  ⁢                                                                          ⁢                  used                  ⁢                                                                          ⁢                                      =                                    ⁢                                      >                                                                                                                        odd                  ⁢                                                                          ⁢                  numbered                  ⁢                                                                          ⁢                  columns                  ⁢                                                                          ⁢                  from                  ⁢                                                                          ⁢                  first                  ⁢                                                                          ⁢                  antenna                  ⁢                                                                          ⁢                  and                  ⁢                                                                          ⁢                  even                  ⁢                                                                          ⁢                  numbered                  ⁢                                                                          ⁢                  columns                  ⁢                                                                          ⁢                  for                  ⁢                                                                          ⁢                  second                  ⁢                                                                          ⁢                  antenna                  ⁢                                                                          ⁢                  ar                                                                                            =                ⁢                                            [                                                                                                                  e                        0                                                  (                          1                          )                                                                    ⁢                                              f                                                  1                          ,                          1                                                                                                                                                                        e                        0                                                  (                          1                          )                                                                    ⁢                                              f                                                  1                          ,                          3                                                                                                                          …                                                                                                      e                        0                                                  (                          1                          )                                                                    ⁢                                              f                                                  1                          ,                                                      K                            -                            1                                                                                                                                                                                                                                  e                        1                                                  (                          1                          )                                                                    ⁢                                              f                                                  2                          ,                          1                                                                                                                                                                        e                        1                                                  (                          1                          )                                                                    ⁢                                              f                                                  2                          ,                          3                                                                                                                          …                                                                                                      e                        1                                                  (                          1                          )                                                                    ⁢                                              f                                                  2                          ,                                                      K                            -                            1                                                                                                                                                                                    ⋮                                                        ⋮                                                        ⋱                                                        ⋮                                                                                                                                      e                                                  K                          -                          1                                                                          (                          1                          )                                                                    ⁢                                              f                                                  K                          ,                          1                                                                                                                                                                        e                                                  K                          -                          1                                                                          (                          1                          )                                                                    ⁢                                              f                                                  K                          ,                          3                                                                                                                          …                                                                                                      e                                                  K                          -                          1                                                                          (                          1                          )                                                                    ⁢                                              f                                                  K                          ,                                                      K                            -                            1                                                                                                                                                          ]                        ⁡                          [                                                                                          s                      0                                              (                        1                        )                                                                                                                                                        s                      2                                              (                        1                        )                                                                                                                                  ⋮                                                                                                              s                                              K                        -                        2                                                                    (                        1                        )                                                                                                        ]                                +                    ⁢                                    [                                                                                                                  e                        0                                                  (                          2                          )                                                                    ⁢                                              f                                                  1                          ,                          2                                                                                                                                                                        e                        0                                                  (                          2                          )                                                                    ⁢                                              f                                                  1                          ,                          4                                                                                                                          …                                                                                                      e                        0                                                  (                          2                          )                                                                    ⁢                                              f                                                  1                          ,                          K                                                                                                                                                                                                        e                        1                                                  (                          2                          )                                                                    ⁢                                              f                                                  2                          ,                          2                                                                                                                                                                        e                        1                                                  (                          2                          )                                                                    ⁢                                              f                                                  2                          ,                          4                                                                                                                          …                                                                                                      e                        1                                                  (                          2                          )                                                                    ⁢                                              f                                                  2                          ,                          K                                                                                                                                                          ⋮                                                        ⋮                                                        ⋱                                                        ⋮                                                                                                                                      e                                                  K                          -                          1                                                                          (                          2                          )                                                                    ⁢                                              f                                                  K                          ,                          2                                                                                                                                                                        e                                                  K                          -                          1                                                                          (                          2                          )                                                                    ⁢                                              f                                                  K                          ,                          4                                                                                                                          …                                                                                                      e                                                  K                          -                          1                                                                          (                          2                          )                                                                    ⁢                                              f                                                  K                          ,                          K                                                                                                                                ]                        ⁡                          [                                                                                          s                      1                                              (                        2                        )                                                                                                                                                        s                      3                                              (                        2                        )                                                                                                                                  ⋮                                                                                                              s                                              K                        -                        1                                                                    (                        2                        )                                                                                                        ]                                                              =                ⁢                              [                                                                                                      e                      0                                              (                        1                        )                                                              ⁢                                          f                                              1                        ,                        1                                                                                                                                                        e                      0                                              (                        2                        )                                                              ⁢                                          f                                              1                        ,                        2                                                                                                                                                        e                      0                                              (                        1                        )                                                              ⁢                                          f                                              1                        ,                        3                                                                                                                                                        e                      0                                              (                        2                        )                                                              ⁢                                          f                                              1                        ,                        4                                                                                                              …                                                                                            e                      0                                              (                        1                        )                                                              ⁢                                          f                                              1                        ,                                                  K                          -                          1                                                                                                                                                                                e                      0                                              (                        2                        )                                                              ⁢                                          f                                              1                        ,                        K                                                                                                                                                                                    e                      1                                              (                        1                        )                                                              ⁢                                          f                                              2                        ,                        1                                                                                                                                                        e                      1                                              (                        2                        )                                                              ⁢                                          f                                              2                        ,                        2                                                                                                                                                        e                      1                                              (                        1                        )                                                              ⁢                                          f                                              2                        ,                        3                                                                                                                                                        e                      1                                              (                        2                        )                                                              ⁢                                          f                                              2                        ,                        4                                                                                                              …                                                                                            e                      1                                              (                        1                        )                                                              ⁢                                          f                                              2                        ,                                                  K                          -                          1                                                                                                                                                                                e                      1                                              (                        2                        )                                                              ⁢                                          f                                              2                        ,                        K                                                                                                                                          ⋮                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          ⋮                                                                                                                        e                                              K                        -                        1                                                                    (                        1                        )                                                              ⁢                                          f                                              K                        ,                        1                                                                                                                                                        e                                              K                        -                        1                                                                    (                        2                        )                                                              ⁢                                          f                                              K                        ,                        2                                                                                                                                                        e                                              K                        -                        1                                                                    (                        1                        )                                                              ⁢                                          f                                              K                        ,                        3                                                                                                                                                        e                                              K                        -                        1                                                                    (                        2                        )                                                              ⁢                                          f                                              K                        ,                        4                                                                                                              …                                                                                            e                                              K                        -                        1                                                                    (                        1                        )                                                              ⁢                                          f                                              K                        ,                                                  K                          -                          1                                                                                                                                                                                e                                              K                        -                        1                                                                    (                        2                        )                                                              ⁢                                          f                                              K                        ,                        K                                                                                                                  ]                    ⁡                      [                                                                                s                    0                                          (                      1                      )                                                                                                                                        s                    1                                          (                      2                      )                                                                                                                                        s                    2                                          (                      1                      )                                                                                                                                        s                    3                                          (                      2                      )                                                                                                                    ⋮                                                                                                  s                                          K                      -                      2                                                              (                      1                      )                                                                                                                                        s                                          K                      -                      1                                                              (                      2                      )                                                                                            ]                              
The above expression includes columns that are alternating between the transmit antennas. This cannot be formulated as the product of a block diagonal matrix and a Fast Fourier Transform (FFT) matrix unless the channels from both transmit antennas are equal for all subcarriers. Thus, only in such a rare special case would the receiver decompose into frequency domain equalization followed by Inverse Fast Fourier Transform (IFFT). The last equation is consistent with equation (4).